Q16. - Algebra 1

Question

MULTIPLE CHOICE Antonia is carpeting two of the rooms in her house. The dimensions are shown. What is the total area to be carpeted?

F $x^2+\mathbf{3}x$

G $\mathbf{2}{x}^2+\mathbf{6}x-\mathbf{10}$

H $x^2+\mathbf{3}x-\mathbf{5}$

J $\mathbf{8}x+\mathbf{12}$

Step-by-Step Solution

Answer

The total area to be carpeted is $\mathbf{2}{x}^2+\mathbf{6}x-\mathbf{10}$.

Step 1. Write formula of area of rectangle.

Area of rectangle is given by $A=\text{length }\left(l\right)\times \text{breadth }\left(b\right)$.

Step 2. Find the area of first room.

Replace $l$ with $x+3$ and b with x, in $A=\text{length }\left(l\right)\times \text{breadth }\left(b\right)$.

$$\begin{gathered} A_1=\left(x+3\right)x \\ =x^2+3x \end{gathered}$$

Step 3. Find the area of second room.

Replace $l$ with $x+5$ and b with $x-2$, in $A=\text{length }\left(l\right)\times \text{breadth }\left(b\right)$.

$$\begin{gathered} A_2=\left(x+5\right)\left(x-2\right) \\ =x\left(x-2\right)+5\left(x-2\right) \\ =x^2-2x+5x-10 \\ =x^2+3x-10 \end{gathered}$$

Step 4. Find the total area.

To find the total area, add $A_1$ and $A_2$.

$$\begin{gathered} A_1+A_2=\left(x^2+3x\right)+\left(x^2+3x-10\right) \\ =2x^2+6x-10 \end{gathered}$$ 

Therefore, the total area to be carpeted is $2x^2+6x-10$. As the obtained area completely matches with option G. Therefore, option G is the correct answer.